The easiest way to check is via google (look up the mfg's website and see what they say, usually they'll have a place to download drivers by OS if they interface with the OS at all).
But very little in the way of hardware is going to care about 64bit v 32bit OS - you may need OS-specific drivers for some items so you can check for their availability but that's about it. Assuming you're buying new parts (not pulling something you've had on the shelf for 3 years) the only place I can imagine running into that sort of problem is the free crapware that comes with many HDDs and DVD Burners (e.g. backup software provided might be out of date) but if that actually happens, just don't use the crapware.
If you're really worried, then by all means google away. But at least make the task less onerous by eliminating the obvious items like the case and monitor.